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            <h1 style="display: none">第四章 二重积分和多元函数</h1>
            
            
              <div class="markdown-body">
                
                <h2 id="二重积分"><a href="#二重积分" class="headerlink" title="二重积分"></a>二重积分</h2><h3 id="一、二重积分概念和性质"><a href="#一、二重积分概念和性质" class="headerlink" title="一、二重积分概念和性质"></a>一、二重积分概念和性质</h3><h6 id="性质"><a href="#性质" class="headerlink" title="性质"></a>性质</h6><p><img src="https://i0.hdslb.com/bfs/album/262072760dfd2046089e91193098de5d86651a38.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/05dbaff894fbaee06d2e204d6a8e23db6bf91a9f.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/7d62409d5baa768136e993fa59f91e4a2e04e83d.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/295ef262086669a41af82d86d123dc243bd56376.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="考点"><a href="#考点" class="headerlink" title="考点"></a>考点</h6><p><img src="https://i0.hdslb.com/bfs/album/1079301ee7d35569e531843836021b3a2fda460f.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/c9d26be2afe804fe4bcfaa76b11da36337465ea3.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题"><a href="#真题" class="headerlink" title="真题"></a>真题</h6><p>2009</p>
<p><img src="https://i0.hdslb.com/bfs/album/a7734d98108fd7ad8e83cfc0d2bd7f44894d2a5b.webp" srcset="/img/loading.gif" lazyload></p>
<p>2016</p>
<p><img src="https://i0.hdslb.com/bfs/album/38a7de4abb7f8c55629098b4ac20dced7b2bb9ce.webp" srcset="/img/loading.gif" lazyload></p>
<p>2019</p>
<p><img src="https://i0.hdslb.com/bfs/album/0502550f9c85901ad2ded3a400a081556f317106.webp" srcset="/img/loading.gif" lazyload></p>
<p>2020</p>
<p><img src="https://i0.hdslb.com/bfs/album/b49ed0dfaa3673d73b00c382a3cc479f28dfa1c2.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="二、二重积分比较大小"><a href="#二、二重积分比较大小" class="headerlink" title="二、二重积分比较大小"></a>二、二重积分比较大小</h3><p><img src="https://i0.hdslb.com/bfs/album/3c83877bdf7c3e67b58b76527afc20190d49ef74.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题"><a href="#例题" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/2ebe436a88282f38f0dd95b16a6576274d544df0.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/605ae66b2df82189f44b32113c9ab30d580bcace.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="估值定理"><a href="#估值定理" class="headerlink" title="估值定理"></a>估值定理</h6><p><img src="https://i0.hdslb.com/bfs/album/67627b1353de12d090f253087da44588f88c8225.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="三、二重积分直角坐标系下的计算公式"><a href="#三、二重积分直角坐标系下的计算公式" class="headerlink" title="三、二重积分直角坐标系下的计算公式"></a>三、二重积分直角坐标系下的计算公式</h3><h6 id="公式"><a href="#公式" class="headerlink" title="公式"></a>公式</h6><p><img src="https://i0.hdslb.com/bfs/album/e9846e9e856d88b5a8fa09aede2ab3b41ff4862b.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="X型图"><a href="#X型图" class="headerlink" title="X型图"></a>X型图</h6><p><img src="https://i0.hdslb.com/bfs/album/3191d29191ae8595d3573ad57c13a3fcecb6bd64.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="Y型图"><a href="#Y型图" class="headerlink" title="Y型图"></a>Y型图</h6><p><img src="https://i0.hdslb.com/bfs/album/c8c1a801e28bdee9ea5fdee1584f129ede4da071.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="求解过程"><a href="#求解过程" class="headerlink" title="求解过程"></a>求解过程</h6><p><img src="https://i0.hdslb.com/bfs/album/b262b253694d8b5b37f5d7a2348fcd1f09261fa5.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-1"><a href="#例题-1" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/ee2779a8e55235a606210126ceab7ce33f1af048.webp" srcset="/img/loading.gif" lazyload alt="例题一"></p>
<p><img src="https://i0.hdslb.com/bfs/album/7267fbc0235bed2adb47e360ffd5ce9f2b385044.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/0faf2972e0b1103bf55b4d01abc7a37c44571f39.webp" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<p><img src="https://i0.hdslb.com/bfs/album/6411b05f373c2b7aa97cddf1a82159928f7232be.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="四、超越积分顺序选择"><a href="#四、超越积分顺序选择" class="headerlink" title="四、超越积分顺序选择"></a>四、超越积分顺序选择</h3><h6 id="类型"><a href="#类型" class="headerlink" title="类型"></a>类型</h6><p><img src="https://i0.hdslb.com/bfs/album/3474e2259ae4c0635b6dee2079f60dcca9f073a0.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-2"><a href="#例题-2" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/0890aa3fc42c3a6421b33ef254384d2831cfdd88.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/ac392aefbffd075fded68ba1918e653b0304d5f2.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/fa366c943de2ae732fd3af4de6229fdb20f407c1.webp" srcset="/img/loading.gif" lazyload alt="例题二"></p>
<p><img src="https://i0.hdslb.com/bfs/album/e51d79e533931e4d293a04651e482ec20f850445.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="五、交换积分次序"><a href="#五、交换积分次序" class="headerlink" title="五、交换积分次序"></a>五、交换积分次序</h3><p><img src="https://i0.hdslb.com/bfs/album/d401ade030bbebf2087f2462629d359d56b72d8a.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-1"><a href="#真题-1" class="headerlink" title="真题"></a>真题</h6><p>2010</p>
<p><img src="https://i0.hdslb.com/bfs/album/20c0fc798d408523f46eff63d001e41948c247ca.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/6815cfc84a62d1b5b8c44eb4e71d803b6de4e2dd.webp" srcset="/img/loading.gif" lazyload></p>
<p>2016</p>
<p><img src="https://i0.hdslb.com/bfs/album/7382f5704d22883d378f50e35667b67a1a40d310.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/84b411053d0727eedc5aba29ddd01fd318301106.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/264a6f7a7e2520c35bba207710605237e6992783.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/0d849886d0d2ece8ec3cbafe9dd2e3554969eec7.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="六、极坐标下二重积分计算"><a href="#六、极坐标下二重积分计算" class="headerlink" title="六、极坐标下二重积分计算"></a>六、极坐标下二重积分计算</h3><h6 id="计算步骤"><a href="#计算步骤" class="headerlink" title="计算步骤"></a>计算步骤</h6><p><img src="https://i0.hdslb.com/bfs/album/2116a928aa3fb5a46a9c2fac253591ce4b5a44b6.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/bfc578a21dc9eab7115bc3702de0aa04464c1c21.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/7c8c4ac829c8d45f1ec09e43b311b30a31b16fb3.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="常见的积分图像"><a href="#常见的积分图像" class="headerlink" title="常见的积分图像"></a>常见的积分图像</h6><p>第一种</p>
<p><img src="https://i0.hdslb.com/bfs/album/cf63edf12da120be79f94efb25c9d8462d8404ed.webp" srcset="/img/loading.gif" lazyload></p>
<p>第二种</p>
<p><img src="https://i0.hdslb.com/bfs/album/81ba2afa1bcb3c53215f8cc23399d082b239c799.webp" srcset="/img/loading.gif" lazyload></p>
<p>第三种</p>
<p><img src="https://i0.hdslb.com/bfs/album/e50b06865219afcab4f7fa7522e9f2f0782240cc.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-2"><a href="#真题-2" class="headerlink" title="真题"></a>真题</h6><p>2014</p>
<p><img src="https://i0.hdslb.com/bfs/album/334bd2de76c502270a6e4390b7d6f37f4904bf37.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/6af50a8ff0b6e1085ce9fd6e4dcfe012fadd763d.webp" srcset="/img/loading.gif" lazyload></p>
<p>2017</p>
<p><img src="https://i0.hdslb.com/bfs/album/6edfcc856ca45fd6d4f15c806a7a77129fb2523b.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/fbddb2db6ba7cff8875a62fdff1250d527970486.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/759b05dbd578bfb5e240a67e8840f0c5d454f45a.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="七、二重积分对称性"><a href="#七、二重积分对称性" class="headerlink" title="七、二重积分对称性"></a>七、二重积分对称性</h3><h6 id="结论"><a href="#结论" class="headerlink" title="结论"></a>结论</h6><p><img src="https://i0.hdslb.com/bfs/album/068692e66ac0367ce2326c565807daa2b84ec5fa.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="二重积分的偶倍奇零"><a href="#二重积分的偶倍奇零" class="headerlink" title="二重积分的偶倍奇零"></a>二重积分的偶倍奇零</h6><p><img src="https://i0.hdslb.com/bfs/album/361e3faca0ae879297ca773d5dc795b836e8c8fb.webp" srcset="/img/loading.gif" lazyload></p>
<p>2016</p>
<p><img src="https://i0.hdslb.com/bfs/album/3029f069f8456e9e739586137bf6aba943e2703c.webp" srcset="/img/loading.gif" lazyload></p>
<p>2015</p>
<p><img src="https://i0.hdslb.com/bfs/album/62ae4f09af47a81ce2763a22adf6f7b1142d8091.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/9733f41f8668c1a5557ed7f1aafbbef50bb146ed.webp" srcset="/img/loading.gif" lazyload></p>
<h2 id="多元函数微分学"><a href="#多元函数微分学" class="headerlink" title="多元函数微分学"></a>多元函数微分学</h2><h3 id="九、多元函数定义域"><a href="#九、多元函数定义域" class="headerlink" title="九、多元函数定义域"></a>九、多元函数定义域</h3><h6 id="了解"><a href="#了解" class="headerlink" title="了解"></a>了解</h6><p><img src="https://i0.hdslb.com/bfs/album/f74af98d65096654b06e58d254f4f7d653b125a0.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-3"><a href="#真题-3" class="headerlink" title="真题"></a>真题</h6><p>2009</p>
<p><img src="https://i0.hdslb.com/bfs/album/8879a583b94c14819955620213c60711ff5b6323.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="十、二元函数对应法则"><a href="#十、二元函数对应法则" class="headerlink" title="十、二元函数对应法则"></a>十、二元函数对应法则</h3><p>解法：</p>
<ol>
<li>换元</li>
<li>配凑</li>
</ol>
<p><img src="https://i0.hdslb.com/bfs/album/c7c234f26e80d58e01aeb7e3d0b1bc34933eb10d.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="十一、二元函数极限"><a href="#十一、二元函数极限" class="headerlink" title="十一、二元函数极限"></a>十一、二元函数极限</h3><h6 id="了解-1"><a href="#了解-1" class="headerlink" title="了解"></a>了解</h6><p><img src="https://i0.hdslb.com/bfs/album/078d2c6f119c6d3fad853f8ebdbc43f46d84ca83.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-3"><a href="#例题-3" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/c998992c172e985521e62d949a8749cc4af11d2e.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="十二、一阶偏导概念"><a href="#十二、一阶偏导概念" class="headerlink" title="十二、一阶偏导概念"></a>十二、一阶偏导概念</h3><h6 id="概念"><a href="#概念" class="headerlink" title="概念"></a>概念</h6><p><img src="https://i0.hdslb.com/bfs/album/41cc7a30692527a233e1cb1a545e2c4792c3c62f.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/a6195e010960c627932aa844ee33cb34e8b6b43c.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-4"><a href="#真题-4" class="headerlink" title="真题"></a>真题</h6><p>2016</p>
<p><img src="https://i0.hdslb.com/bfs/album/0016c2b8164ead3c08aa13269a2cb3505fca7047.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-4"><a href="#例题-4" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/0acdc46f173dda14c801c0cda4ca07e8543e9461.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/5201046f02390212d376bb47255b9738c47edb0c.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="十三、全微分"><a href="#十三、全微分" class="headerlink" title="十三、全微分"></a>十三、全微分</h3><h6 id="公式-1"><a href="#公式-1" class="headerlink" title="公式"></a>公式</h6><p><img src="https://i0.hdslb.com/bfs/album/c35210edb44c750c5639499b28493713566d163f.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-5"><a href="#例题-5" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/bd0451cf90bdd33d3f32a57351f7672d7593edaa.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/c83a5bc5a740942704d4e890b1d860f8814cad6a.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-5"><a href="#真题-5" class="headerlink" title="真题"></a>真题</h6><p><img src="https://i0.hdslb.com/bfs/album/56f3ae75bc9034445d0d8ab533900ffeb7cde98a.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="十四、可微与偏导的关系"><a href="#十四、可微与偏导的关系" class="headerlink" title="十四、可微与偏导的关系"></a>十四、可微与偏导的关系</h3><h6 id="关系"><a href="#关系" class="headerlink" title="关系"></a>关系</h6><p><img src="https://i0.hdslb.com/bfs/album/671cfb2f4334b23a3fd5bacc64aa1de3592a0721.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="十五、可微的本质"><a href="#十五、可微的本质" class="headerlink" title="十五、可微的本质"></a>十五、可微的本质</h3><h6 id="概念-1"><a href="#概念-1" class="headerlink" title="概念"></a>概念</h6><p><img src="https://i0.hdslb.com/bfs/album/105e1a699c042b6bc702947f33479263d11a31ff.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/29b37ee7144f3b15b1588c6e10fd7fab68f347ef.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-6"><a href="#例题-6" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/b75e57fdee580550ed027f0212f0f45d8d2fc070.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/920c5c8faa3ef761d99aef4e9ce33385588faecd.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="十六、二阶偏导"><a href="#十六、二阶偏导" class="headerlink" title="十六、二阶偏导"></a>十六、二阶偏导</h3><h6 id="理解"><a href="#理解" class="headerlink" title="理解"></a>理解</h6><p><img src="https://i0.hdslb.com/bfs/album/ed0fa7e8ea6e0477172857e1c186210d32fab23b.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/16f785b467b0df3b391ad5c1736f5feb4fc3f594.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="例题-7"><a href="#例题-7" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/a5ca121fbf8c71fcec3ecf3df8a90fd2054339db.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-6"><a href="#真题-6" class="headerlink" title="真题"></a>真题</h6><p>2018</p>
<p><img src="https://i0.hdslb.com/bfs/album/adfc73a3c3a252597acfde53f706e9fbedaec3fd.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="十七、二元隐函数一阶偏导"><a href="#十七、二元隐函数一阶偏导" class="headerlink" title="十七、二元隐函数一阶偏导"></a>十七、二元隐函数一阶偏导</h3><p>概念</p>
<p><img src="https://i0.hdslb.com/bfs/album/da86e869aa94b276b8dfea4a3b88e3f61620615c.webp" srcset="/img/loading.gif" lazyload alt="概念"></p>
<h6 id="例题-8"><a href="#例题-8" class="headerlink" title="例题"></a>例题</h6><p><img src="https://i0.hdslb.com/bfs/album/e73a713269c918c92bfe4f1af2bb33a08fec2c73.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-7"><a href="#真题-7" class="headerlink" title="真题"></a>真题</h6><p>2015</p>
<p><img src="https://i0.hdslb.com/bfs/album/6894792d290cd160ff4065646a0536730a7422e5.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/54d98b878caf8fcf16afad3646182d1da0bb9877.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="十八、二元隐函数二阶偏导"><a href="#十八、二元隐函数二阶偏导" class="headerlink" title="十八、二元隐函数二阶偏导"></a>十八、二元隐函数二阶偏导</h3><h6 id="概念-2"><a href="#概念-2" class="headerlink" title="概念"></a>概念</h6><p><img src="https://i0.hdslb.com/bfs/album/0295107638f33616b2ffcb73be89725b1710f903.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="真题-8"><a href="#真题-8" class="headerlink" title="真题"></a>真题</h6><p>2017</p>
<p><img src="https://i0.hdslb.com/bfs/album/6cb6926d5b44b36f36d3cb57aacbad2b6da2134a.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/b41b137ceda5d19f7ec57a2af9af1c907cc8f4fc.webp" srcset="/img/loading.gif" lazyload></p>
<h3 id="十九、多元复合函数"><a href="#十九、多元复合函数" class="headerlink" title="十九、多元复合函数"></a>十九、多元复合函数</h3><h6 id="链式法则"><a href="#链式法则" class="headerlink" title="链式法则"></a>链式法则</h6><p><img src="https://i0.hdslb.com/bfs/album/47c18ed231ae497d184962deae33c81191dbc19a.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="具体多元复合函数求导"><a href="#具体多元复合函数求导" class="headerlink" title="具体多元复合函数求导"></a>具体多元复合函数求导</h6><p>2016</p>
<p><img src="https://i0.hdslb.com/bfs/album/b7d2b878f78e5496b3658cf9fa6c4bcece79d74e.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="抽象复合函数求导"><a href="#抽象复合函数求导" class="headerlink" title="抽象复合函数求导"></a>抽象复合函数求导</h6><p>题型</p>
<p><img src="https://i0.hdslb.com/bfs/album/bc7dd886813ba8bc2e48ff90f7d57247a795f08a.webp" srcset="/img/loading.gif" lazyload></p>
<p>例题</p>
<p><img src="https://i0.hdslb.com/bfs/album/bba71fac880042cbe9a7ed1f338ce5c35701bfab.webp" srcset="/img/loading.gif" lazyload></p>
<p>真题</p>
<p>2014</p>
<p><img src="https://i0.hdslb.com/bfs/album/a6c2c3f90976d4a27c4ede2e56b71240e0e0c4fb.webp" srcset="/img/loading.gif" lazyload></p>
<p>2016</p>
<p><img src="https://i0.hdslb.com/bfs/album/34390518dcab17273a04be238fb9682421a2a53e.webp" srcset="/img/loading.gif" lazyload></p>
<h6 id="隐函数求导易错题"><a href="#隐函数求导易错题" class="headerlink" title="隐函数求导易错题"></a>隐函数求导易错题</h6><p><img src="https://i0.hdslb.com/bfs/album/6f2e94d29fe0c74b9a001119b48aca2b3b8c87b6.webp" srcset="/img/loading.gif" lazyload></p>
<p><img src="https://i0.hdslb.com/bfs/album/ba5cb4bc91c34ec93823461c3ce3bba618141d3b.webp" srcset="/img/loading.gif" lazyload></p>

                
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